Average drift speed of charge carriers + ratios + resistances. Need help?

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Average drift speed of charge carriers + ratios + resistances. Need help?

A $12 V$ $12V$ battery with negligible internal resistance is connected to two resistors $Y$ $Y$ and $Z$ $Z$ , the resistors are connected in series.
The resistors are made from wires of the same material. The wire $Y$ $Y$ has a diameter $d$ $d$ and length $l$ $l$ . The wire $Z$ $Z$ has a diameter $2 d$ $2d$ and length $2 l$ $2l$ .

$(i)$ $(i)$ Determine the ratio:
average drift speed of the charge carriers in $Y$ $Y$ $/$ $/$ average drift speed of the charge carriers in $Z$ $Z$ .

$(i i)$ $(ii)$ Show that:
resistance of $Y$ $Y$ $/$ $/$ resistance of $Z$ $Z$ $= 2$ $= 2$ .

$(i i i)$ $(iii)$ Determine the potential difference across $Y$ $Y$ .

$(i v)$ $(iv)$ Determine the ratio:
power dissipated in $Y$ $Y$ $/$ $/$ power dissipated in $Z$ $Z$ .

P.S. sorry for long question... but it has to be done...

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Answer 1 · 2017-09-19T02:40:30

I got

Explanation:

$(i)$ $(i)$ The average drift velocity $u$ $u$ is given by the expression

$u = \frac{I}{n A q}$ $u = I /(n A q)$
where $I$ $I$ is the current flowing in the resistor, $n$ $n$ is the number density of charge-carrier, $A$ $A$ is cross sectional area of the conductor, and $q$ $q$ is the charge on the charge-carrier, in this case electronic charge.
$:.u_Y/u_Z=(I /(n A q))_Y/(I /(n A q))_Z$

As the resistors are made from the same wire $=>nand q$ are same for both, are connected in series $=>$ current flowing in the resistors is same. Above ratio reduces to

$u_Y/u_Z= A _Z/A_Y$ .......(1)
$=>u_Y/u_Z= ((pid_Z^2)/4)/((pid_Y^2)/4)$

Inserting given values we get

$u_Y/u_Z= ((2d)^2)/d^2$
$=>u_Y/u_Z= 4$

$(ii)$ Resistance $R$ of a conducting wire is given as

$R=(rhoL)/A$
where $rho$ is resistivity of material, $L$ length of wire and $A$ ist area of cross section.

Now required ratio

$R_Y/R_Z=((rhoL)/A)_Y/((rhoL)/A)_Z$

Noting that $rho$ is same for both resistors,

$R_Y/R_Z=L_Y/L_Z A_Z/A_Y$

Using (1) and inserting given values we get

$R_Y/R_Z=l/(2l)xx 4$
$=>R_Y/R_Z=2$
Proved

$(iii)$ Total resistance in the circuit

$R_T=R_Y+R_Z$
$=>R_T=R_Y+R_Y/2=3/2R_Y$ .....(2)

From Ohm's law, Current in the circuit

$I=V/R_T$

Potential difference across resistor $Y$ is give as

$V_Y=IR_Y$
$V_Y=(V/R_T)R_Y$

Using (2) we get

$V_Y=(V/(3/2R_Y))R_Y$
$=>V_Y=2/3V$

Inserting value of voltage source we get

$V_Y=2/3xx12=8V$

$(iv)$ Power dissipated in resistor is given as

$P=I^2R$

The required ratio

$P_Y/P_Z=(I^2R)_Y/(I^2R)_Z$
$=>P_Y/P_Z=R_Y/R_Z=2$ , (given in $(ii)$ above)

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Average drift speed of charge carriers + ratios + resistances. Need help?

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